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Agenda Introduction to trigonometry- Right-angled triangles, theta, etc. Trigonometric functions Angle of elevation Angle of depression Applicability in real life Simple problems involving angles of elevation/depression Introduction to Trigonometry Formed from Greek words 'trigonon' (triangle) and 'metron' (measure). Trigonometric triangles are always rightangled triangles More on Trigonometry A branch of mathematics that studies • Triangles • Relationship between sides and angles between sides Uses • Describes relationship trigonometric between sides/angles functions Sides of a Right-angled Triangle Hypotenuse • Opposite to the right-angle • Longest side Adjacent • Side that touches θ Opposite • Side opposite to θ Theta 8th letter of the Greek alphabet Represented by “θ” A variable, not a constant Commonly used in trigonometry to represent angle values Trigonometric Functions Sin (Sine)= ratio of opposite side to the hypotenuse Cos (Cosine)= ratio of adjacent side to the hypotenuse Tan (Tangent)= ratio of opposite side to the adjacent side Easier way to remember Sin, Tan, Cos SOH CAH TOA TOA: Tangent = Opposite ÷ Adjacent (T=O/A) CAH: Cosine = Adjacent ÷ Hypotenuse (C=A/H) SOH: Sine = Opposite ÷ Hypotenuse (S=O/H) Angle of Elevation The angle of elevation is the angle between the horizontal line and the observer’s line of sight, where the object is above the observer Angle of Depression The angle of depression is the angle between the horizontal line and the observer’s line of sight, where the object is below the observer Applicability of Angles of Elevation and Depression Used by architects to design buildings by setting dimensions Used by astronomers for locating apparent positions of celestial objects Used in computer graphics by designing 3D effects properly Used in nautical navigations by sailors (sextants) Many other uses in our daily lives To find sides If given two sides of the right triangle, you can use the Pythagorean Theorem 𝑎2 + 𝑏2 = 𝑐 2 If given one side and one angle, you will use a trigonometric function. Finding Angles To find angles, you have to use the inverse of the trigonometric ratios. Sin => sin−1 Cos => cos −1 Tan => tan−1 Example 1 • Question: You see a huge tree that is 50 feet in height and it casts a shadow of length 60 feet. You are standing at the tip of the shadow. What is the degree of elevation from the end of the shadow to the top of the tree? Example 2 From a point 115 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 64.3 degrees . Find the height of the tree to the nearest foot. Example 3 From a point 10 feet from the base of a flag pole, the angle of elevation to the top of the flag pole is 67.4 degrees . Find the height of the flag pole to the nearest foot. Example 4 If the distance from a helicopter to a tower is 300 feet and the angle of depression is 40.2 degrees , find the distance on the ground from a point directly below the helicopter to the tower Example 5 The angle of depression of one side of a lake, measured from a balloon 2500 feet above the lake is 43 degrees . The angle of depression to the opposite side of the lake is 27 degrees . Find the width of the lake Overall summary • • • • Draw the diagram Identify the known values Form equations Solve We hope you have enjoyed our presentation Thank you for your kind attention! Please ask reasonable questions, if any.